Richard P. Stanley Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139

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Parking Functions and Noncrossing Partitions
Richard P. Stanley
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA 02139
[email protected]
Submitted: August 12, 1996; Accepted: November 12, 1996
Dedicated to Herb Wilf on the occasion of his sixty- fth birthday
Abstract
A parking function is a sequence (a1 ; : : : ; an ) of positive integers such that
if b1 b2 bn is the increasing rearrangement of a1 ; : : : ; an , then bi i.
A noncrossing partition of the set [n] = f1; 2; : : : ; ng is a partition of the set
[n] with the property that if a < b < c < d and some block B of contains
both a and c, while some block B 0 of contains both b and d, then B =
B 0. We establish some connections between parking functions and noncrossing
partitions. A generating function for the ag f -vector of the lattice NCn+1 of
noncrossing partitions of [n + 1] is shown to coincide (up to the involution !
on symmetric function) with Haiman's parking function symmetric function.
We construct an edge labeling of NCn+1 whose chain labels are the set of all
parking functions of length n. This leads to a local action of the symmetric
group Sn on NCn+1 .
MR primary subject number: 06A07
MR secondary subject numbers: 05A15, 05E05, 05E10, 05E25
1. Introduction. A parking function is a sequence (a1 ; : : : ; an) of positive
integers such that if b1 b2 bn is the increasing rearrangement of
a1 ; : : : ; an, then bi i.1 Parking functions were introduced by Konheim and
Weiss [14] in connection with a hashing problem (though the term \hashing"
was not used). See this reference for the reason (formulated in a way which
Minor variations of this de nition appear in the literature, but they are equivalent to the de nition given here. For instance, in [31] parking functions are obtained from the de nition given here
by subtracting one from each coordinate.
1
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now would be considered politically incorrect) for the terminology \parking
function." Parking functions were subsequently related to labelled trees and
to hyperplane arrangements. For further information on these connections see
[31] and the references given there. In this paper we will develop a connection
between parking functions and another topic, viz., noncrossing partitions.
A noncrossing partition of the set [n] = f1; 2; : : : ; ng is a partition of
the set [n] (as de ned e.g. in [29, p. 33]) with the property that if a < b <
c < d and some block B of contains both a and c, while some block B 0 of
contains both b and d, then B = B 0 . The study of noncrossing partitions
goes back at least to H. W. Becker [1], where they are called \planar rhyme
schemes." The systematic study of noncrossing partitions began with Kreweras
[15] and Poupard [22]. For some further work on noncrossing partitions, see
[5][21][25][28] and the references given there.
A fundamental property of the set of noncrossing partitions of [n] is that it
can be given a natural partial ordering. Namely, we de ne if every block
of is contained in a block of . In other words, is a re nement of . Thus
the poset NCn of all noncrossing partitions of [n] is an induced subposet of the
lattice n of all partitions of [n] [29, Example 3.10.4]. In fact, NCn is a lattice
with a number of remarkable properties. We will develop additional properties
of the lattice NCn which connect it directly with parking functions.
2. The parking function symmetric function. Let P be a nite
graded poset of rank n with ^0 and ^1 and with rank function . (See [29,
Ch. 3] for poset terminology and notation used here.) Let S be a subset of
[n 1] = f1; 2; : : : ; n 1g, and de ne P (S ) to be the number of chains ^0 =
t0 < t1 < < ts = ^1 of P such that S = f(t1 ); (t2 ); : : : ; (ts 1 )g. The
function P is called the ag f -vector of P . For S [n 1] further de ne
X
( 1)jS T j P (T ):
P (S ) =
T S
The function P is called the ag h-vector of P . Knowing
knowing P since
X
P (S ) =
P (T ):
P
is the same as
T S
For further information on ag f -vectors and h-vectors (using a di erent terminology), see [29, Ch. 3.12].
There is a kind of generating function for the ag h-vector which is often
useful in understanding the combinatorics of P . Regarding n as xed, let
S [n 1] and de ne a formal power series QS = QS (x) = QS (x1 ; x2 ; : : :) in
the (commuting) indeterminates x1 ; x2 ; : : : by
X
xi xi xin :
QS =
i1 i2
ij <i
j +1
in
j S
1
2
if
2
QS is known as Gessel's quasisymmetric function [10] (see also [16, x5.4][18][24,
Ch. 9.4]). The functions QS , where S ranges over all subsets of [n 1], are
linearly independent over any eld. For our ranked poset P we then de ne
X
FP =
P (S )QS :
S [n
1]
2
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3
This de nition (in a di erent but equivalent form) was rst suggested by R.
Ehrenborg [6, Def. 4.1] and is further investigated in [30]. One of the results
of [30] (Thm. 1.4) is the following proposition (which is equivalent to a simple
generalization of [29, Exercise 3.65]).
2.1 Proposition. Let P be as above. If every interval [u; v] of P is ranksymmetric (i.e., [u; v] has as many elements of rank i as of corank i), then FP
is a symmetric function of x1 ; x2 ; : : :.
We now consider the case P = NCn+1 . (We take NCn+1 rather than NCn
because NCn+1 has rank n.) It is well-known that every interval in NCn+1 is
self-dual and hence rank-symmetric. (This follows from the fact that NCn+1
is itself self-dual [15, x3][27, Thm. 1.1] and that every interval of NCn+1 is a
product of NCi 's [21, x1.3].) Hence FNCn is a symmetric function, and we
can ask whether it is already known. In fact, FNCn has previously appeared
in connection with parking functions, as stated below in Theorem 2.3. First we
provide some background information related to parking functions.
Let Pn denote the set of all parking functions of length n. The symmetric
group Sn acts on Pn by permuting coordinates. Let PFn = PFn (x) denote the
Frobenius characteristic of the character of this action [17, Ch. 1.7]. Thus if
+1
+1
PFn =
X
`n
;ns
is the expansion of PFn in terms of Schur functions, then ;n is the multiplicity
of the irreducible character of Sn indexed by in the action of Sn on Pn . The
symmetric function PFn was rst considered in the context of parking functions
by Haiman [13, xx2.6 and 4.1]. Following Haiman, we will give a formula for
PFn from which its expansion in terms of various symmetric function bases
is immediate. The key observation (due to Pollak [8, p. 13] and repeated in
[13, p. 28]) is the following (which we state in a slightly di erent form than
Pollak). Let Zn+1 denote the set f1; 2; : : : ; n + 1g, with addition modulo n + 1.
Then every coset of the subgroup H of Znn+1 generated by (1; 1; : : : ; 1) contains
exactly one parking function. From this it follows easily that
(1)
PFn = n +1 1 [tn ]H (t)n+1 ;
where [tn ]G(t) denotes the coecient of tn in the power series G(t), and where
H (t) = 1 + h1 t + h2 t2 + = (1 x t)(11 x t) ;
1
2
the generating function for the complete symmetric functions hi . (Throughout
this paper we adhere to symmetric function terminology and notation as in
Macdonald [17].)
The following proposition summarizes some of the properties of PFn which
follow easily from equation (1).
2.2 Proposition. (a) We have the following expansions.
PFn =
X
`n
(n + 1)`() 1 z 1 p
(2)
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=
X
1
n + 1 s (1
`n
s
n+1 )
"
(3)
#
1 Y i + n m
=
n+1 i
n
`n
X n(n 1) (n `() + 2)
=
h :
m
1 ()! mn ()!
`n
"
#
Y n + 1
X 1
m:
=
n + 1 i i
`n
X
!PFn
4
(4)
(5)
(6)
Here s (1n+1 ) denotes s with n +1 variables set equal to 1 and the others to 0,
and is evaluated explicitly e.g. in [17, Example 4, p. 45]. Moreover, `() is the
number of parts of ; z is as in [17, p. 24]; mi () denotes the number of parts
of equal to i; and ! is the standard involution [17, pp. 21{22] on symmetric
functions.
(b) We also have that
X
n0
PFn tn+1 = (tE ( t))h 1i ;
(7)
P
where E (t) = n0 en tn , en denotes the nth elementary symmetric function,
and h 1i denotes compositionalQinverse.
Proof. (a) Let C (x; y) = i;j (1 xiyj ) 1, the well-known \Cauchy product." Then H (t)n+1 is obtained by setting n + 1 of the yi's equal to t and
the others equal to 0. From this all the expansions in (a) follow from (1) and
well-known expansions for C (x; y) and !x C (x; y) (where !x denotes ! acting
on the x variables only). To give just one example (needed in the rst proof of
Theorem 2.3), we have
!xC (x; y) =
Hence
!PFn =
X
X
1
m (x)e (y):
e (1
n
+
1
`n
m :
n+1 )
Equation (6) now follows from the simple fact that ek (1n+1 ) = n+1
k . We
should point out that (2) appears (in a dual form) in [11, (9)], (3) appears in
[13, (28)], and (5) appears (again in dual form) in [13, (82)][17, Example 24(a),
p. 35]. A q-analogue of PFn and of much of our Proposition 2.2 appears in [9].
(b) This is an immediate consequence of (1), the fact that
1 = E ( t);
(8)
H (t )
and the Lagrange inversion formula, as in [13, x4.1]. See also [17, Examples
2.24{2.25, pp. 35{36]. 2
Let P be a Cohen-Macaulay poset with ^0 and ^1 such that every interval is
rank-symmetric. Thus FP is a symmetric function. In [30, Conj. 2.3] it was
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5
conjectured that FP is Schur positive, i.e., a nonnegative linear combination of
Schur functions. Equation (3) con rms this conjecture in the case P = NCn+1 .
However, it turns out that the conjecture is in fact false. A counterexample
is provided by the following poset P . The elements of P consist of all integer
vectors (a1 ; a2 ; b1 ; b2 ; b3 ; b4 ) such that 0 a1 5, 0 a2 1, 0 b1 3,
0 b2 ; b3 ; b4 1, and a1 + a2 = b1 + b2 + b3 + b4 , ordered componentwise. It
can be shown that P is lexicographically shellable and hence Cohen-Macaulay,
and it is easy to see that P is locally rank-symmetric (even locally self-dual).
Moreover,
FP = s6 +7s51 +6s42 +2s33 +18s411 +10s321 s222 +20s3111 +5s2211 +8s21111 :
The symmetric functions PFn also have an unexpected connection with the
multiplication of conjugacy classes in the symmetric group (the work of FarahatHigman [7]). For further details see [11][17, Ch. I, Example 7.25, pp. 132{134].
This connection was exploited by Goulden and Jackson [11] to compute some
connection coecients for the symmetric group.
The expansion (5) of PFn in terms of the h 's has a simple interpretation in
terms of parking functions. Suppose that a = (a1 ; : : : ; an ) 2 Pn . Let r1 ; : : : ; rk
be the positive multiplicities of the elements of the multiset fa1 ; : : : ; an g (so
r1 + + rk = n). Then the action of Sn on the orbit Sn a has characteristic
hr hrk . For instance, a set of orbit representatives in the case n = 3 is
(1; 1; 1); (2; 1; 1); (3; 1; 1); (2; 2; 1), and (3; 2; 1). Hence PF3 = h3 + h2 h1 + h2 h1 +
h2 h1 + h31 = h3 + 3h21 + h111 . In general it follows that the coecient q of h
in PFn is equal to the number of orbits of parking functions of length n such
that the terms of their elements have multiplicities 1 ; 2 ; : : : (in some order).
Equation (5) then gives an explicit formula for this number. The total number
of parking functions whose terms
have multiplicities 1 ; 2 ; : : : is q times the
size of the orbit, i.e., q ;n ;::: .
We are now ready to discuss the connection between PFn and noncrossing
partitions. The basic result is the following.
2.3 Theorem. For any n 0 we have
1
1
2
FNCn = !PFn :
+1
Proof. Let = ( ; : : : ; ` ) be a partition of n with ` > 0. It is immediate
1
from the dePnition of FP in Section 1 (see [30, Prop. 1.1]) that if FP is symmetric
and FP = c m , then
c =
1 ; 1 + 2 ; : : : ; 1 + 2 + + ` 1 ):
P(
(9)
The proof now follows by comparing equation (6) with the evaluation of NCn (S )
due to Edelman [4, Thm. 3.2]. 2
It follows from the above discussion that PFn encodes in a simple way the
ag f -vector and ag h-vector of NCn+1 , viz., (1) the coecient of QS in
the expansion of PFn in terms of Gessel's quasisymmetric function is equal to
(S ), and (2) if the elements of S [n 1] are j1 < < jr and if is
NCn
the partition whose parts are the numbers j1 ; j2 j1 ; j3 j2 ; : : : ; n jr , then
the coecient of m in the expansion of PFn in terms of monomial symmetric
+1
+1
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6
functions is equal to NCn (S ). There is a further statistic on NCn+1 closely
related to PFn , namely, the number of noncrossing partitions of [n + 1] of type
, i.e., with block sizes 1 ; 2 ; : : :.
2.4 Proposition. Let be a partition of n. The coecient of h in the
expansion of PFn in terms of complete symmetric functions is equal to the
number u of noncrossing partitions of type .
First proof. Compare equation (5) with the explicit value of the number
of noncrossing partitions of type found by Kreweras [15, Thm. 4]. 2
Second proof. Our second proof is based on the following noncrossing
analogue of the exponential formula due to Speicher [28, p. 616]. (For a more
general result, see [21].) Given a function f : N ! R (where R is a commutative
ring with identity) with f (0) = 1, de ne a function g : N ! R by g(0) = 1 and
+1
X
g(n) =
Let
=fB1 ;:::;Bk g2NCn
P
F (t) = n0 f (n)tn. Then
X
n0
g(n)t
n+1
f (#B1 ) f (#Bk ):
t
= F (t)
h
1
i
:
(10)
(11)
In equation
(10) take f (n) = hn , the complete symmetric function. Then g(n)
P
becomes `n u h . But Proposition 2.2(b), together with equations (11) and
(8), shows that g(n) = PFn , and the proof follows. 2
Note the curious fact that Theorem 2.3 refers to NCn+1 , while Proposition 2.4 refers to NCn. Proposition 2.4, together with the de nition of PFn ,
show that the number of noncrossing partitions of type ` n is equal to the
number of Sn -orbits of parking functions of length n and part multiplicities
. It is easy to give a bijective proof of this fact (shown to me by R. Simion),
which we omit.
3. An edge labeling of the noncrossing partition lattice. If P is a
locally nite poset, then an edge of P is a pair (u; v) 2 P P such that v covers
u (i.e., u < v and no element t satis es u < t < v). An edge labeling of P is
a map : E (P ) ! Z, where E (P ) is the set of edges of P . Edge labelings of
posets have many applications; in particular, if P has what is known as an ELlabeling, then P is lexicographically shellable and hence Cohen-Macaulay [2][3].
An EL-labeling of NCn+1 was de ned by Bjorner [2, Example 2.9] and further
exploited by Edelman and Simion [5]. Here we de ne a new labeling, which
up to an unimportant reindexing is EL and is intimately related to parking
functions.
Let (; ) be an edge of NCn+1 . Thus is obtained from by merging
together two blocks B and B 0 . Suppose that min B < min B 0 , where min S
denotes the minimum element of a nite set S of integers. De ne
(; ) = maxfi 2 B : i < B 0 g;
(12)
where i < B 0 denotes that i is less than every element of B 0 . For instance,
if B = f2; 4; 5; 15; 17g and B 0 = f7; 10; 12; 13g, then (; ) = 5. Note that
(; ) always exists since min B < B 0.
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The labeling of the edges of NCn+1 extends in a natural (and well-known)
way to a labeling of the maximal chains. Namely, if m : ^0 = 0 < 1 < <
n = ^1 is a maximal chain of NCn+1, then set
(m) = ((0 ; 1 ); (1 ; 2 ); : : : ; (n 1 ; n )):
3.1 Theorem. The labels (m) of the maximal chains of NCn+1 consist of
the parking functions of length n, each occuring once.
Proof. If (j ; j+1) = i, then the block of j+1 containing i also contains
an element k > i. Hence the number of j for which (j ; j +1 ) = i cannot
exceed n + 1 i, from which it follows that (m) is a parking function.
Suppose that m and m0 are maximal chains of NCn+1 for which (m) =
(m0 ). We will prove by induction on n that m = m0 . The assertion is clear
for n = 0. Assume true for n 1. Let the elements of m be ^0 = 0 < 1 <
< n = ^1. Suppose that (m) = (a1 ; : : : ; an ). Let r = maxfai : 1 i ng,
and let s = maxfi : ai = rg. We claim that one of the blocks of s 1 is just
the singleton set fr + 1g. If r and r + 1 are in the same block of s 1 , then
we can't have (s 1 ; s ) = r, contradicting as = r. Hence r and r + 1 are in
di erent blocks of s 1. If the block B of s 1 containing r + 1 contained some
element t < r, then by the noncrossing property and the fact that as = r we
have that B is merged with the block B1 of s 1 containing r to get s . But
min B t < r 2 B1 , contradicting as = r. Hence every element of B is greater
than r. If B contained some element t > r + 1, then (since r + 1 = min B ) we
would have ak = r + 1 for some k < r, contradicting maximality of r. This
proves the claim.
We next claim that s is obtained from s 1 by merging the block B1
containing r with the block fr + 1g. Otherwise (since as = r) s is obtained by
merging B1 with some block B2 all of whose elements are greater than r + 1.
For some t > s we must obtain t from t 1 by merging the block B3 containing
r + 1 with the block B4 containing r. Now B3 can't contain an element less
than r + 1 by the noncrossing property of s 1 (since B4 contains both r and
an element greater than r + 1). It follows that (t 1 ; t ) = r, contradicting
the maximality of s and proving the claim.
It is now clear by induction that the chain m can be uniquely recovered
from the parking function (m) = (a1 ; : : : ; an ). Namely, let a0 be the sequence
obtained from (m) by removing as . Then a0 is a parking function of length
n 1. By induction there is a unique maximal chain m : ^0 = 0 < 1 <
< n 1 = ^1 of NCn such that (m ) = a0 . By the discussion above we can
then obtain m uniquely from m by (1) replacing each element i > r of the
ambient set [n] with i + 1, (2) adjoining a singleton block fr + 1g to each i
for i s 1, (3) inserting between s 1 and s a new element obtained from
s 1 by merging the block containing r with the singleton block fr + 1g, and
(4) for i > s adjoining the element r + 1 to the block of i containing r. Hence
we have shown that if (m) = (m0 ), then m = m0 . But it is known [15, Cor.
5.2][4, Cor. 3.3] that NCn+1 has (n + 1)n 1 maximal chains, which is just the
number of parking functions of length n [14, Lemma 1 and x6][8]. Thus every
parking function of length n occurs exactly once among the sequences (m),
and the proof is complete. 2
7
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The above proof of the injectivity of the map from maximal chains to
parking functions is reminiscent of the proof [20, p. 5] that the Prufer code of
a labelled tree determines the tree. Our proof \cheated" by using the fact that
the number of maximal chains is the number of parking functions. We only
gave a direct proof of the injectivity of . However, our proof actually suces
to show also surjectivity since the argument of the above paragraph is valid for
any parking function, the key point being that removing an occurrence of the
largest element of a parking function preserves the property of being a parking
function.
If we de ne a new labeling of NCn+1 by
(; ) = jj (; );
where jj is the number of blocks of , then it is easy to check (using the fact
that every interval of NCn+1 is a product of NCi 's) that every interval [; ]
has a unique maximal chain m : = 0 < 1 < < j = such that
(0 ; 1 ) (1 ; 2 ) (k 1 ; k ):
In other words, is an R-labeling in the sense of [29, Def. 3.13.1]. Moreover,
this maximal chain m has the lexicographically least label (m) of any maximal
chain of the interval [; ]. Thus is in fact an EL-labeling, as de ned in [2,
Def. 2.2] (though there it is called just an \L-labeling."). For the signi cance
of the EL-labeling property, see the rst paragraph of this section. Here we will
just be concerned with the weaker R-labeling property.
De ne the descent set D(a) of a parking function a = (a1 ; : : : ; an ) by
D(a) = fi : ai > ai+1 g:
From the fact that is an R-labeling and [29, Thm. 3.13.2], we obtain the
following proposition.
3.2 Proposition. (a) Let S [n 1]. The number of parking functions a
of length n satisfying D(a) = S is equal to NCn ([n 1] S ).
(b) Let S [n 1]. The number of parking functions a of length n satisfying
D(a) S is equal to NCn ([n 1] S ). This number is given explicitly by
[4, Thm. 3.2] or by equations (4) and (9).
The labeling is closely related to a bijection between the maximal chains
of NCn+1 and labelled trees, di erent from the earlier bijection of Edelman
[4, Cor. 3.3]. Let m : ^0 = 0 < 1 < < n = ^1 be a maximal chain of
NCn+1 . De ne a graph m on the vertex set [n +1] as follows. There will be an
edge ei for each 1 i n. Suppose that i is obtained from i 1 by merging
blocks B and B 0 with min B < min B 0. Then the vertices of ei are de ned to
be (i 1 ; i ) and min B 0 . It is easy to see that m is a tree. Root m at the
vertex 1 and erase the vertex labels. If vi is the vertex of ei farthest from the
root, then move the label i of the edge ei from ei to the vertex vi . Label the
root with 0 and unroot the tree. We obtain a labelled tree Tm on n +1 vertices,
and one can easily check that the map m 7! Tm is a bijection between maximal
chains of NCn+1 and labelled trees on n + 1 vertices.
4. A local action of the symmetric group. Suppose that P is a graded
poset of rank n with ^0 and ^1 such that FP is a symmetric function. If FP
+1
+1
8
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9
is Schur positive, then it is the Frobenius characteristic of a representation
of Sn whose dimension is the number of maximal chains of P . Thus we can
ask whether there is some \nice" representation of Sn on the vector space VP
(over a eld of characteristic zero) whose basis is the set of maximal chains
of P . This question was discussed in [30, x5]. A \nice" representation should
somehow re ect the poset structure. With this motivation, an action of Sn on
VP is de ned to be local [30, x5] if for every adjacent transposition i = (i; i +1)
and every maximal chain
m : ^0 = t0 < t1 < < tn = ^1;
(13)
we have that i (m) is a linear combination of maximal chains of the form
t0 < t1 < < ti 1 < t0i < ti+1 < < tn, i.e., of maximal chains which agree
with m except possibly at ti .
Now let P = NCn+1 . Every interval [; ] of NCn+1 of length two contains
either two or three elements in its middle level. In the latter case, there are three
blocks B1 ; B2 ; B3 of such that is obtained from by merging B1 ; B2 ; B3
into a single block. Moreover, any two of these blocks can be merged to form a
noncrossing partition. Let ij be the noncrossing partition obtained by merging
Bi and Bj , so that the middle elements of the interval [; ] are 12 ; 13 ; 23 .
Exactly one of these partitions ij will have the property that (; ij ) =
(ij ; ), where is de ned by (12). Let us call this partition ij the special
element of the interval [; ]. Now de ne linear transformations i0 : VNCn !
VNCn , 1 i n 1 as follows. Let m be a maximal chain of NCn+1 with
elements ^0 = 0 < 1 < < n = ^1.
Case 1. The interval [i 1 ; i+1 ] contains exactly two middle elements i
and i0 . Then set i0 (m) = m0 , where m0 is given by 0 < 1 < < i 1 <
i0 < i+1 < < n.
Case 2. The interval [i 1 ; i+1 ] contains exactly three middle elements, of
which i is special. Then set i0 (m) = m.
Case 3. The interval [i 1 ; i+1 ] contains exactly three middle elements
i ; i0 , and i00 , of which i00 is special. Then set i0 (m) = m0, where m0 is given
by 0 < 1 < < i 1 < i0 < i+1 < < n .
4.1 Proposition. The action of each i0 on VNCn de ned above yields
a local action of Sn on VNCn . Equivalently, there is a homomorphism ' :
Sn ! GL(VNCn ) satisfying '(i ) = i0 . The Frobenius characteristic of this
action is given by PFn .
Proof. Each maximal chain m corresponds to a parking function (m) via
Theorem 3.1. Thus the natural action of Sn on Pn de ned in Section 2 may
be \transferred" to an action of Sn on the set of maximal chains of NCn+1 .
It is easy to check that and ' agree on the i 's, and the proof follows. 2
The action ' does not quite have the property mentioned at the beginning of
this section that its characteristic is FNCn . By Theorem 2.3, the characteristic
is actually !FNCn . However, we only have to multiply ' by the sign character
(equivalently, de ne a new action '0 by '0 (i ) = '(i )) to get the desired
property.
It is rather surprising that the simple \local" de nition we have given of
' de nes an action of Sn. Perhaps it would be interesting to look for some
+1
+1
+1
+1
+1
+1
+1
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10
more examples. (We need to exclude trivial examples such as w(m) = m for all
w 2 Sn and all maximal chains m.) A few other examples appear in the next
section and in [30, x5]. A further example (the posets of shues of C. Greene
[12]) is discussed in [26] together with the rudiments of a systematic theory of
such actions, but much work needs to be done for a satisfactory understanding
of local Sn -actions.
5. Generalizations. In this section we will brie y discuss two generalizations of what appears above. All proofs are entirely analogous and will be
omitted. Fix an integer k 2 P. A k-divisible noncrossing partition is a noncrossing partition for which every block size is divisible by k. Thus is a
noncrossing partition of a set [kn] for some n 0. Let NC(nk) be the poset of all
k-divisible noncrossing partitions of [kn]. (NC(nk) is actually a join-semilattice
of NCkn . It has ^1 but not a ^0 when k > 1.) The combinatorial properties of
the poset NC(nk) were rst considered by Edelman [4, x4]. If a pair (; ) is an
edge of NC(nk) (i.e., covers in NC(nk) ), then (; ) is an edge of NCn . Hence
the edge-labeling of NCn restricts to an edge-labeling of NC(nk) .
De ne a k-parking function of length n to be a sequence (a1 ; : : : ; an ) of
positive integers such that if b1 b2 bn is the increasing rearrangement
of a1 ; : : : ; an , then bi ki. Let Pn(k) denote the set of all k-parking functions of
length n. The argument of Pollak mentioned in Section 2 that #Pn = (n+1)n 1
easily extends to Pn(k) . Namely, let Zk(n+1) denote the set f1; 2; : : : ; k(n +
1)g with addition modulo k(n + 1). Then every coset of the subgroup H of
Znk(n+1) generated by (1; 1; : : : ; 1) contains exactly k k-parking functions. Hence
#Pn(k) = k(k(n + 1))n 1 . The symmetric group Sn acts on Pn(k) by permuting
coordinates, and we can consider its Frobenius characteristic PF(nk) just as we
did for Pn . The above generalization of Pollak's argument shows that
PF(nk) = n +1 1 [tn ]H (t)k(n+1) :
Proposition 2.2 generalizes straightforwardly to the case of PF(nk) . In particular,
Proposition 2.2(b) takes the form
X
n0
PF(nk) tn+1 = tE ( t)k
h
1
i
:
Theorem 3.1 generalizes as follows.
5.1 Theorem. The labels (m) of the maximal chains of NC(nk+1) consist of
the k-parking functions of length n, each occuring once.
Proposition 3.2 requires some modi cation when extended to k-parking
)
functions because the posets NC(nk+1
do not have a ^0 when k > 1. For these
posets we regard the minimal elements as having rank 0, and we de ne NCnk (S )
and NCnk (S ) for S f0; 1; : : : ; n 1g. Thus for instance NCnk (;) =
(k )
k (;) = 1, and NC k (0) is the number of minimal elements of NCn+1 .
NCn
n
Write [0; n 1] = f0; 1; : : : ; n 1g.
( )
+1
( )
+1
( )
+1
( )
+1
( )
+1
the electronic journal of combinatorics 3 (1996), #R20
11
5.2 Proposition. (a) Let S [n 1]. The number of k-parking functions
a of length n satisfying D(a) = S is equal to
k
n
( )
NC +1
([n 1] S ) +
k
n
( )
NC +1
([0; n 1] S ):
(b) Let S [n 1]. The number of k-parking functions a of length n
satisfying D(a) S is equal to NC k ([0; n 1] S ). This number is given
n
explicitly by [4, Thm. 4.2].
Note, however, that there does not seem to be a nice generalization of
Theorem 2.3. The quasisymmetric function FNCnk is not a symmetric function
when k > 1, and we know of no simple connection between the ag f -vector
)
of NC(nk+1
and the symmetric function PF(nk) , nor between the number of kdivisible noncrossing partitions of a given type and PF(nk) .
)
Proposition 4.1 extends straightforwardly to NC(nk+1
. The natural action of
(k )
Sn on Pn is transferred via Theorem 5.1 to an action on VNC k . This action
n
is a permutation representation on the maximal chains, and is readily seen to
be local. Its characteristic is PF(nk) .
There is a di erent generalization of noncrossing partitions due to Reiner
[23] (a special case had earlier appeared in a di erent guise in [19], as explained
in [23]) that we have not looked at very closely. Reiner regards ordinary noncrossing partitions as corresponding to the root system An and constructs analogues for the root systems Bn and Dn . Actually, for every subset S [n] he
constructs a lattice NCBD
n (S ) interpolating between the Bn analogue (the case
S = ;) and the Dn analogue (the case S = [n]). The lattices NCBD
n (S ) do not
always have self-dual intervals [23, Remark on p. 13], but at least every interval
is rank-symmetric. Thus by Proposition 2.1, this implies that NCBD
n (S ) is a
symmetric function. This symmetric function only depends on the cardinality
s of S , so we write FnBD (s) for FNCBD
. We also write FnB for FnBD (0).
(S )
n
Let [n]n denote the set of all sequences (a1 ; : : : ; an ) of positive integers with
ai n. We call such a sequence a Bn-parking function, for the following reason.
Let PFBn be the Frobenius characteristic of the action of Sn on [n]n obtained
by permuting coordinates. It follows from [23, Prop. 7] that
( )
+1
( )
+1
( )
+1
FnB = !PFBn :
Thus in analogy with Theorem 2.3 it makes sense to think of the elements of
[n]n as Bn -parking functions. Reiner's result [23, Prop. 7] makes it easy to give
an analogue of Proposition 2.2. Let us simply mention the formula
PFBn = [tn ]H (t)n ;
from which the analogues of all parts of Proposition 2.2 follow easily. In particular, the analogue of Proposition 2.2(b) takes the form
X
n1
h 1i
n
PFBn tn = log (tE ( tt)) :
(14)
the electronic journal of combinatorics 3 (1996), #R20
Comparing Proposition 2.2 with equation (14) yields the curious result that
exp
X
n1
PFBn tn =
n
X
n0
PFn tn :
What is missing from the analogy between An and Bn noncrossing partitions
is the analogue of Theorem 3.1, i.e., a labeling of NCBn+1 such that the labels of
the maximal chains are the Bn -parking functions. We have not looked at this
question and recommend it as an interesting open problem.
For the general case of NCBD
n (S ), it follows from [23, Thm. 11] that
FnBD (s) = FnB s PF0n ;
where PF0n is the Frobenius characteristic of the action of Sn on all sequences
(a1 ; : : : ; an ) of positive integers whose increasing rearrangement b1 bn
satis es b1 = 1 and bi i 1 for 2 i n. We have not considered further
properties of the symmetric function FnBD (s).
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